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The minimum of the heights of the prime ideals containing the ideal. The height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468301.png" /> of a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468302.png" /> in a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468303.png" /> is the largest number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468304.png" /> (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468305.png" /> if such a number does not exist) such that there exists a chain of different prime ideals
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468306.png" /></td> </tr></table>
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The co-height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468307.png" /> of a prime ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468308.png" /> is defined as the largest <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h0468309.png" /> for which there exists a chain of prime ideals
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The minimum of the heights of the prime ideals containing the ideal. The height $  \mathop{\rm ht} ( \mathfrak p ) $
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of a prime ideal $  \mathfrak p $
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in a ring  $  A $
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is the largest number  $  h $
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(or  $  \infty $ if such a number does not exist) such that there exists a chain of different prime ideals
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683010.png" /></td> </tr></table>
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$$
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\mathfrak p _ {0}  \subset  \mathfrak p _ {1}  \subset  \dots \subset  \
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\mathfrak p _ {h}  = \mathfrak p .
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$$
  
In other words,
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The co-height  $  \mathop{\rm coht} ( \mathfrak p ) $
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of a prime ideal  $  \mathfrak p $
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is defined as the largest  $  h $
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for which there exists a chain of prime ideals
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683011.png" /></td> </tr></table>
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$$
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\mathfrak p  = \
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\mathfrak p _ {0}  \subset  \mathfrak p _ {1}  \subset  \dots \subset  \mathfrak p _ {h}  \neq  A.
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$$
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Equivalently,
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$$
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\mathop{\rm ht} ( \mathfrak p )  =  \mathop{\rm dim} ( A _ {\mathfrak p }  ),\ \
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\mathop{\rm coht} ( \mathfrak p )  =  \mathop{\rm dim} ( A / \mathfrak p ),
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$$
  
 
where dim denotes the dimension of the corresponding Krull ring. The height of a prime ideal is equal to the codimension of the variety defined by the ideal, while the co-height equals the dimension of this variety. The height and the co-height of a prime ideal are connected by the inequality
 
where dim denotes the dimension of the corresponding Krull ring. The height of a prime ideal is equal to the codimension of the variety defined by the ideal, while the co-height equals the dimension of this variety. The height and the co-height of a prime ideal are connected by the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683012.png" /></td> </tr></table>
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$$
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\mathop{\rm ht} ( \mathfrak p ) +
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\mathop{\rm coht} ( \mathfrak p )
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\leq    \mathop{\rm dim}  A,
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$$
  
which becomes an equality if, for example, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683013.png" /> is a local [[Cohen–Macaulay ring|Cohen–Macaulay ring]].
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which becomes an equality if, for example, $  A $
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is a local [[Cohen–Macaulay ring|Cohen–Macaulay ring]].
  
The prime ideals of height zero are the minimal prime ideals. The existence of prime ideals of height one in Noetherian integral domains is established by the principal ideal theorem: The height of a non-zero principal ideal is one (cf. [[Krull ring|Krull ring]]). A more general result — Krull's theorem — interconnects the height with the number of generators of the ideal: In a Noetherian ring the height of an ideal generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683014.png" /> elements is not larger than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683015.png" />, and conversely: A prime ideal of height <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683016.png" /> is the smallest of all prime ideals containing some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046830/h04683017.png" /> elements. In particular, any ideal in a Noetherian ring has finite height; this is not true of the co-height [[#References|[2]]].
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The prime ideals of height zero are the minimal prime ideals. The existence of prime ideals of height one in Noetherian integral domains is established by the principal ideal theorem: The height of a non-zero principal ideal is one (cf. [[Krull ring|Krull ring]]). A more general result — Krull's theorem — interconnects the height with the number of generators of the ideal: In a Noetherian ring the height of an ideal generated by $  r $
 +
elements is not larger than $  r $,  
 +
and conversely: A prime ideal of height $  r $
 +
is the smallest of all prime ideals containing some $  r $
 +
elements. In particular, any ideal in a Noetherian ring has finite height; this is not true of the co-height [[#References|[2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Krull,  "Primidealketten in allgemeinen Ringbereichen" , Berlin-Leipzig  (1928)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Nagata,  "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.-P. Serre,  "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer  (1965)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  W. Krull,  "Primidealketten in allgemeinen Ringbereichen" , Berlin-Leipzig  (1928)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Nagata,  "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  O. Zariski,  P. Samuel,  "Commutative algebra" , '''1''' , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.-P. Serre,  "Algèbre locale. Multiplicités" , ''Lect. notes in math.'' , '''11''' , Springer  (1965)</TD></TR></table>

Latest revision as of 17:43, 11 January 2021


The minimum of the heights of the prime ideals containing the ideal. The height $ \mathop{\rm ht} ( \mathfrak p ) $ of a prime ideal $ \mathfrak p $ in a ring $ A $ is the largest number $ h $ (or $ \infty $ if such a number does not exist) such that there exists a chain of different prime ideals

$$ \mathfrak p _ {0} \subset \mathfrak p _ {1} \subset \dots \subset \ \mathfrak p _ {h} = \mathfrak p . $$

The co-height $ \mathop{\rm coht} ( \mathfrak p ) $ of a prime ideal $ \mathfrak p $ is defined as the largest $ h $ for which there exists a chain of prime ideals

$$ \mathfrak p = \ \mathfrak p _ {0} \subset \mathfrak p _ {1} \subset \dots \subset \mathfrak p _ {h} \neq A. $$

Equivalently,

$$ \mathop{\rm ht} ( \mathfrak p ) = \mathop{\rm dim} ( A _ {\mathfrak p } ),\ \ \mathop{\rm coht} ( \mathfrak p ) = \mathop{\rm dim} ( A / \mathfrak p ), $$

where dim denotes the dimension of the corresponding Krull ring. The height of a prime ideal is equal to the codimension of the variety defined by the ideal, while the co-height equals the dimension of this variety. The height and the co-height of a prime ideal are connected by the inequality

$$ \mathop{\rm ht} ( \mathfrak p ) + \mathop{\rm coht} ( \mathfrak p ) \leq \mathop{\rm dim} A, $$

which becomes an equality if, for example, $ A $ is a local Cohen–Macaulay ring.

The prime ideals of height zero are the minimal prime ideals. The existence of prime ideals of height one in Noetherian integral domains is established by the principal ideal theorem: The height of a non-zero principal ideal is one (cf. Krull ring). A more general result — Krull's theorem — interconnects the height with the number of generators of the ideal: In a Noetherian ring the height of an ideal generated by $ r $ elements is not larger than $ r $, and conversely: A prime ideal of height $ r $ is the smallest of all prime ideals containing some $ r $ elements. In particular, any ideal in a Noetherian ring has finite height; this is not true of the co-height [2].

References

[1] W. Krull, "Primidealketten in allgemeinen Ringbereichen" , Berlin-Leipzig (1928)
[2] M. Nagata, "Local rings" , Interscience (1962)
[3] O. Zariski, P. Samuel, "Commutative algebra" , 1 , Springer (1975)
[4] J.-P. Serre, "Algèbre locale. Multiplicités" , Lect. notes in math. , 11 , Springer (1965)
How to Cite This Entry:
Height of an ideal. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Height_of_an_ideal&oldid=14422
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article